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ON THE LUBRICATING AND OTHER PROPERTIES OF THIN OILY FILMS.
[Philosophical Magazine, Vol. xxxv. pp. 157—162, 1918.]
THE experiments about to be described were undertaken to examine more particularly a fact well known in most households. A cup of tea, standing in a dry saucer, is apt to slip about in an awkward manner, for which a remedy is found in the introduction of a few drops of water, or tea, wetting the parts in contact. The explanation is not obvious, and I remember discussing the question with Kelvin many years ago, with but little progress.
It is true that a drop of liquid between two curved surfaces draws them together and so may increase the friction. If d be the distance between the plates at the edge of the film, T the capillary tension, and a the angle of contact, the whole force is*
ZATcosa vm • —•-,--......+BTsma,
A being the area of the film between the plates and B its circumference. If the fluid wets the plate, a = 0 and we have simply 2AT/d. For example, if d = 6 x 10~5 cm., equal to a wave-length of ordinary light, and T(&s for water) be 74 dynes per cm., the force per sq. cm. is 25 x 10B dynes, a suction of 2^ atmospheres. For the present purpose we may express d in terms of the radius of curvature (p) of one of the surfaces, the other being supposed flat, and the distance (as) from the centre to the edge of the film. In two dimensions d = x?l2,p, and A (per unit of length in the third dimension) =* 2#, so that the force per unit of length is SpT/%, inversely as x. On the other hand, in the more important case of symmetry round the common normal A = TTXZ, and the whole force is 4>7rpT, independent of x, but increasing with the radius of curvature. For example, if T=74 dynes per cm., and (0 = 100 cm., the force is 925 dynes, or the weight of about 1 gramf. The radius of curvature (p) might of course be much greater. There are circumstances where this force is of importance; but, as we shall see presently, it does not avail to explain the effects .now under consideration.
* See for example Maxwell on Capillarity. Collected Papers, Vol. n. p. 571. [t This result does not correspond to the stated values of T and p, which imply a force of 93,000 dynes, or the -weight of about 95 grammes. W. F. S.]f the second and third degrees respectively in Cj/Cg, and lead to very complicated expressions. Without, however, including the oblique line, it may be shown that the two lines h=hi from x=Q to clt and h — hz=lchi ^°^ GI to Cj + Cg, with 2/i2=3JT, as on pp. 530, 531, make F/P a minimum when h^h^, "provided 4/c3-8fc2 + fc-3<};0, leading to fc <j: 2-06 approximately: since these lines have been shown on pp. 530, 531 to make P a maximum, they therefore also make F a minimum when k Jf. 2-06. With 7c=2-06, (44) and (39) make jyP=4-0137i3/c, P=-19469Mtfc2/7ia2. But, with F/P positive, the minimum value of (44) occurs at 7c = 2, when P/P=47i1/c, P=0-2/*D'c2//i12. Accordingly, as this value of F/P is not a minimum for all variations outside the straight line h=h1, the actual minimum value of F/P must be less than 47^/c. W. P. S.]ng there is some light inxxix. p. 128, 1874). I do not know the date of Thoulet's use of the solution, but suspect that it was subsequent to Sonstadt's.